Ulysses’ trip

Mar 3, 2020

Normal vectors of a scalar field

Article cover image with no practical usage This article gives the formula for the normal vectors of a surface defined by a scalar field on $\mathbb R^n$ . The normal vector of the graph of the function $y=f\!\left(\mathbf x\right)$ at $\left(\mathbf x_0,f\!\left(\mathbf x_0\right)\right)$ is $\left(\nabla f\!\left(\mathbf x_0\right),-1\right)$ . This also provides us a way to recover a scalar field from the normal vectors of its graph: normalizing the vectors so that the last component is $-1$ , and then integrate the rest components.

/math #calculus #vector analysis

Feb 29, 2020

It is Feb 29 today!

Article cover image with no practical usage It is Feb 29 today. The date appears once for as long as 4 years!

/update #fooling around

Feb 9, 2020

Monkey-patching graciously

Article cover image with no practical usage Monkey-patching is a powerful tool in programming. In this article, I used techniques of Ruby metaprogramming to define a series of methods def_after, def_before, etc. to help monkey-patching. They look graciously in that we can use it to shorten the codes for monkey-patching (avoiding aliasing and repeating codes).

/programming #ruby #meta programming #long paper

Jan 26, 2020

Amazing Siteleaf

Article cover image with no practical usage I have been using Siteleaf to manage my blog. It is just convenient and amazing.

/update #update #jekyll

Jan 25, 2020

Hyperellipsoids in barycentric coordinates

Article cover image with no practical usage In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.

/math #linear algebra #long paper

Jan 6, 2020

Use complex numbers as canonical variables

Article cover image with no practical usage In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: $\mathbf c\coloneqq\alpha\mathbf q+\mathrm i\beta\mathbf p$ . It turns out that we can write canonical equations $\frac{\mathrm d\mathbf c}{\mathrm dt}=-2\mathrm i\alpha\beta\frac{\partial\mathcal H}{\partial\mathbf c^*}$ , Poisson brackets $\left\{f,g\right\}=-2\mathrm i\alpha\beta \left(\frac{\partial f}{\partial\mathbf c}\cdot \frac{\partial g}{\partial\mathbf c^*}- \frac{\partial f}{\partial\mathbf c^*}\cdot \frac{\partial g}{\partial\mathbf c}\right)$ , and canonical transformations $\frac{\partial\mathbf c^*}{\partial\mathbf c'^*}= \frac{\partial\mathbf c'}{\partial\mathbf c}, \frac{\partial\mathbf c}{\partial\mathbf c'^*}= -\frac{\partial\mathbf c'}{\partial\mathbf c^*}$ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.